Space Matters: Evaluating the Spatial Component of

Variation in New York City School Test Scores

Marc A. Scott

New York University, New York City, USA

Deinya Phenix

New York University, New York City, USA

Benjamin B. Kennedy

Gettysburg College, Gettysburg, PA, USA

Summary. Spatially structured variation between schools has important implications for ad-

ministrative policy and reform. Recent policy initiatives at both national and local levels seek to

ensure that children have access to sufficiently well-functioning schools, regardless of where

they live. The advent of Geographical Information Systems (GIS) has enabled analysts and

policy makers to identify potential “hot spots” – geographically contiguous areas of poor school

performance – particularly in large urban school systems. This may coincide with district ef-

fects, but we often discover patterns within a local district and across district boundaries. We

combine approaches of spatial data analysis with multi-level modeling to formally assess both

global and local spatially-driven variation. Covariate effects are introduced and spatial struc-

ture is reassessed under these controls.

The authors thank Amy Schwartz and Norm Fruchter for providing the data and for theirsupport and guidance. We additionally thank Dana Draghicescu, Mark Handcock, GarySimon and Jeff Simonoff for illuminating discussions on spatial modeling.

1. The importance of geography in educational outcomes

This paper examines geographically structured disparity in public school outcomes. Mostlarge urban school systems with diverse populations and high levels of social inequalityexhibit educational outcome inequity, i.e., profound variation across schools in academicachievement. School-level academic success or failure is an important barometer of neighbor-hood conditions, indicating effective or ineffective teaching and learning within each institu-tion and possibly the absence of committed service providers (Wilson, 1987; Brooks-Gunn,Duncan and Aber, 1997; Jencks and Mayer, 1990; Small and Newman, 2001). Maps ofschool-level performance indicate that variation in performance may have spatial structure,which potentially reflects a host of factors, including residential sorting and concentratedpoverty, compounded by an uneven teacher labor market and administrative competence.

As illustration of this variation, we present in Figure 1 school-level fourth grade readingscores for the New York City public schools in academic year 2003-2004, reported as thepercent of pupils at or above grade level (henceforth, PPE). A thick line separates Brooklynfrom Queens (and given the resolution, Manhattan from The Bronx), with a legend in theleftmost portion of the plot. The 633 schools are depicted as single points–it is at theselocations that schools are situated and to which scores are attributed. Does this figuresuggest spatial structure to school test performance? There seem to be reasonably consistent

2 Scott et al.

regional differences in average school performance; one goal of this paper is to decomposethe variation so that we can quantify the extent to which differences are spatially driven.Also depicted on this map are 32 school districts, which historically had administrativecontrol over the schools within them. These of course capture some coarse informationregarding space, for two schools in the same district are on average more proximate thantwo schools in different districts. Variation both between and within districts is apparent,but less clear is what may be happening on the border between two districts. Do schoolsnear each other, but in different districts have similar outcomes, and does this suggest thatneighborhood factors are playing a role?

[Figure 1 here]

More formally, to what extent do these patterns reflect district performance (a “be-tween” effect), individual school differences within district, and local, spatially patternedvariation that may cross district boundaries? How will this partitioning change as we intro-duce explanatory covariates? In essence, we might say that these outcomes exhibit spatialstructure, but what do we mean when we say this?

Multi-level models (MLM; Bryk and Raudenbush, 2002; Gelman and Hill, 2007) willguide our answers to these questions, but in order to do so, they must be implementedin a manner that reflects coarse and fine levels of spatial variation. Fortunately, coarsespatial variation can often be captured via school district effects. In large, urban schoolsystems school districts are usually organized geographically, and to some extent, thesereflect the neighborhoods from which students are drawn (the variation in the distancetraveled by students to attend each school is an interesting area to study in its own right).In the particular example we use in this paper, we have longitudinal school-level test scores,so unmeasured differences between schools can be identified as school effects in an MLMframework. Some of the local structure that we wish to identify lies somewhere betweendistrict- and school-level variation.

A brief discussion of what local spatial structure represents is warranted. We do thisin the context of educational outcomes, but the ideas apply more generally. Test scoresare likely to be associated with an array of measurable characteristics, both at the stu-dent, teacher, school and district level. In this analysis, we examine school-level aggregateoutcomes and predictors. To the extent that some important predictors are unmeasuredor unmeasurable, there may be unexplained correlation between outcomes for schools withsimilar levels of these predictors. To the extent that an omitted predictor is not randomlydistributed across units that have distinct locations, that predictor may exhibit spatial struc-ture, or the tendency for its levels to be correlated with physically nearby units. Povertyis a good example of a predictor that exhibits spatial structure; neighborhoods changesocio-economically in a relatively smooth fashion, and this yields poorer and more wealthyneighborhoods. In fact, if we redraw figure 1 using a measure of student poverty as anoutcome, it will exhibit spatial structure extremely similar to that of PPE.

We can formally model different components of variation using MLMs – this is calleda variance components analysis (see Searle, Casella and McCulloch, 1992). Our baselinemodel utilizes the following representation:

Yikt = Xiktβ + αi + δk + S(i) + εikt. (1)

The outcome Yikt, for the ith unit (school) in the kth group (district) at time t, is thesum of a mean process captured through covariates with Xiktβ, a school-level effect αi, a

Quantifying Spatial Components 3

district effect δk and an error term εikt. The common assumptions of normality, conditionalindependence and homogeneity of variance will be applied to these effects. The S(i) termreflects the unique contribution based on the location of school i – it is left largely unspecifiedin this formulation. Some analyses of school performance might ignore S(i) altogether,assuming that S(i) = 0, conditional on αi and δk. Some might view it as part of the errorstructure, reflecting a legitimate concern that correct inference relies on proper specificationof the correlation structure between observations. We will discuss three sets of models: thefirst ignores spatial correlation and attempts to assess it through a residual analysis; thesecond captures it through what is known as spatially-lagged variables; the third uses aform of ’local’ random effect. The approaches will be compared and contrasted.

In the first class of models (those that ignore local spatial correlation), we label thecombined errors ζikt = S(i) + εikt and then test whether their estimates, the residuals,exhibit spatial correlation. A significant finding informs our interpretation of stylized maps,and statistical tests are a better guide to spatial correlation than visual assessment. For thesecond class, we write down an explicit model for the S(i) term. Spatial correlation can becaptured in variogram (see Cressie, 1993) and spatial autoregressive models (SAR models;see Haining, 2003, Anselin and Bera, 1998). The former modeling class assumes a parametricform for the correlation between any two errors; for example, corr(εikt, εi′k′t) = exp(−r/d),where r is the distance between the locations specified by i and i′, and d is a scalingparameter. While this model class handles certain forms of spatial autocorrelation, itsassumption of intrinsic stationarity is unlikely to hold in data of the kind we explore in thispaper; relatedly, these models were developed for outcomes that could occur on the entireplane, while school buildings are at fixed locations. An alternative, SAR models, can beimplemented in a straightforward manner as a mean adjustment, and the correspondingregression coefficients reflect the strength of local correlation. Implemented in an MLMframework, SAR models allow us to gauge the impact of spatial information on our variancecomponents, but they do not generate separate variance components of their own. Togenerate a separate, local spatial variance component, we will model S(i) using a variantof grid-based approaches exploiting nearest neighbor structure. This is our third class ofmodels.

In what follows, we employ these three different modeling techniques to assess the im-portance of local spatial variation. The first class will test for complete spatial randomness,and we apply it to the residuals (or random effect estimates) from an MLM fit; failing thistest suggests the presence of “spatial signal.” The second is the SAR model that helps toquantify the role of local spatial information in these outcomes. We finally contrast theseresults to those obtained with a nearest neighbor variance component approach. In eachcase, we evaluate the role of covariates in explaining the variance components. Returningto the motivation for this study, these statistical models and tests will help us understandwhich aspects of structure initially identified through shaded maps of school performanceare most important, and the role of covariates, if any, in explaining that structure. Someof these features have important policy implications, in light of several local and federalinitiatives to provide equal educational opportunities to all children, regardless of wherethey happen to live.

We care about spatial correlation for several reasons. First, it proxies for unobservedcovariates; identifying locally under- or over-performing “hot spots” in the data could sug-gest additional predictors (see Grimson, Wang and Johnson, 1981, for an example fromthe health sciences). Further, a variance components analysis that tracks the reductionor elimination of spatial correlation as we add predictors suggests the ‘level’ (e.g., school,

4 Scott et al.

neighborhood, district) in which effects operate. Lastly, from a policy perspective, spatialstructure could be indicative of administrative neglect, differential resource allocation orvariation in managerial effectiveness. After controlling for some non-manipulable covari-ates, if what remains exhibits no local spatial structure, then a form of adjusted spatialequity has been obtained.

This paper will develop methods that assess the role of proximity in annual Englishproficiency test scores in the New York City (NYC) public schools for the six consecutiveacademic years, beginning in 1998. In section 2, we discuss the data and metrics to be usedin the analysis. We discuss three ways to evaluate and quantify the presence or absenceof local spatial structure in section 3, all based on an underlying MLM framework. Insection 4, we discuss the inclusion of covariates and how these relate to the coarse and localspatial structure. In section 5, we evaluate our models using the three different approaches,applying them to NYC test score data. We find that these scores contain significant spatialcomponents under each of our definitions. In this analysis, we examine the interplay betweenspatial structure and two important predictors, and the implications for variance partitions.In section 6, we discuss the relevance of our findings and the contribution of the spatialinformation to evaluation and policy research.

2. Data and metrics

The school system under study is New York City’s, one of the United States’ largest schoolsystems, with decentralized control taking the form of 32 geographically organized Com-munity School Districts (hereafter, “districts”). These local districts, with 15-50 schools,sometimes span several official and unofficial “neighborhoods” (New York City Departmentof City Planning, 2006).

We use school-level performance and poverty data published by the New York CityDepartment of Education. Specifically, our outcome is the percent of fourth grade studentsscoring at or above grade level on the academic year (AY) state reading examinations foryears beginning 1998-2003. Our poverty measure is the percent of students eligible forfree and reduced cost lunch. We include measures of race and ethnicity that are anchoredin AY2003-2004 – the school demographics with respect to this measure do not changedramatically in the period under study. Other predictors were available, but these twosufficiently illustrate our methodology. Although spatial effects may be present in middleand high schools, this study is limited to elementary school performance for consistency ininterpretation of location. Catchment zones are geographically based for most elementaryschools, while zones for upper grade levels are less regular. Schools’ physical addresseswere geocoded with a 99 percent match rate, resulting in latitude-longitude coordinates fornearly every school.

3. Evaluating local spatial variation

In many datasets of this type, there are some coarse areal units that form natural geographicgroupings, whether they be states, counties, census tracts, or in our case, school districts.Elementary school students tend to go to school within their district and more often to their“local” (or most proximate) school. In this setting, districts manage resources, and thusdistrict performance reflects both managerial and (aggregated) neighborhood effects. Welabel such district effects as coarse spatial components and note that they reflect multiple

Quantifying Spatial Components 5

processes.

Local spatial variation is much harder to define, but we draw on previous work in spatialstatistics and spatial econometrics to do so. If there were no local spatial component, thenafter netting out covariate and coarse spatial effects, we would expect the residuals to beindependent, regardless of location. In particular, nearby values should not be correlated.We formalize this intuition in three different ways, but in each, we use a multi-level modelingframework. This provides us with a clear variance decomposition that we use to gauge therelative impact of local spatial effects and traditional predictors in explaining our outcomes.

Baseline multi-level model

Our baseline multi-level model establishes the coarse spatial, unit-level, and residual varia-tion in the absence of further predictors or local spatial components. We are fortunate tohave repeated measures for schools, so consistent differences at the unit or school level areidentifiable. Our model is thus

Yikt = β0 + β1t + β2t2 + αi + δk + εikt, (2)

with αi ∼ N(0, σ2α), δk ∼ N(0, σ2

δ), and εikt ∼ N(0, σ2ε), independent of one another. The

coarse spatial variance component is given by σ2δ and unit-level variation is given by σ2

α. Thequadratic was sufficiently flexible to capture the mean trend for the six years in our study.The extent to which the variance components, including σ2

ε , change with the addition of alocal spatial component and other predictors will inform our understanding of each of theirroles.

We note that the district and school effects are estimated as random rather than fixedeffects. This specification allows us to assess their relative contribution to the overall vari-ance more directly and to include time constant predictors at the school and district level.We will elaborate on this aspect of the specification when we include predictors in section4.

Test of residual spatial correlation

Our first assessment of local spatial variation is a post-hoc non-parametric statistical testapplied to the residuals, rather than a pure model-based evaluation. Having included coarse,district-level spatial effects in our multi-level model, we draw on the spatial statistics lit-erature for tests that examine the residuals for a lack of “spatial randomness.” To bemeaningfully interpreted in the manner we wish, these residuals must have all covariate (orfixed) effects and all random effects removed. To do this, we compute the best linear unbi-

ased predictions (BLUPs; see Robinson, 1991), α̂i and δ̂k, given the relevant observations(all predictors and outcomes associated with school i in district k) and all parameter esti-mates in these models. For the baseline model, the predictors are simply X = (1, t, t2), and

the parameters estimates are Θ̂ = {β̂, σ̂2α, σ̂2

δ , σ̂2ε}. The BLUPs are given by E(αi|Θ̂, Y, X)

and E(δk|Θ̂, Y, X), where it is understood that only relevant subsets of Y and X will be

used in computing each expectation. The residuals are then ε̂ikt = Yikt− (Xiktβ̂ + α̂i + δ̂k).

We use Geary’s c test for residual spatial autocorrelation, which relies on a weightsmatrix {wuv} that we assume is row normalized to sum to one. Let {e1, e2, . . . , en} bethe n residuals we wish to test – in our application these will be based on BLUPs and are

6 Scott et al.

aggregated over time. Then the Geary coefficient is

c =(n − 1)

∑

u,v wuv(eu − ev)2

2n∑

u(eu − r̄)2, (3)

where r̄ = 1n

∑

u eu (Cliff and Ord, 1981). The weights matrix can be quite general, forexample reflecting the distance between observations. We use a very simple definition: foreach school, the five nearest neighbors are assigned uniform weights, and all remainingschools are given zero weight. We note that this choice of weights clearly reflects localrelationships and importantly, may cross district borders. If nearby residuals are verysimilar, the numerator in the expression for c will be small relative to the denominator.A statistical test of whether c is unusually small can be based on asymptotic theory orthrough a permutation-based procedure. We use the latter, which derives the samplingdistribution of c under the complete spatial randomness assumption – that school locationsare arbitrary. In our application, we take eu to be the median over the six time points foreach school, so our test only picks up consistently spatially correlated residuals. Note thatwe may also wish to examine the random effects estimates, α̂i, for spatial correlation, asthis would reflect another feature of test scores.

SAR model

Our implementation of the simultaneous autoregressive model (SAR) for spatial data buildson the framework used to evaluate spatial structure in the Geary test. Using a weightsmatrix {wil} that reflects the importance of local information, SAR models introduce aspatially-lagged component, a new predictor, Zikt =

∑

l wilYlkt, so that the new modelbecomes

Yikt = β0 + β1t + β2t2 + αi + δk + ρZikt + εikt. (4)

In practice, the weights are chosen so that nearest neighbors are given the most importance;we give equal weight to the five nearest neighbors, and zero weight to the remainder, roughlyanalogous to our setup in the Geary test. The SAR model thus contains an explicit termthat accounts for correlation between outcomes. When ρ is significant, residual spatialautocorrelation is present, and its magnitude can be directly interpreted as the weightone should assign to neighboring schools when trying to predict the current one. In datawith local positive spatial correlation, it will usually be between 0 and 1, though it is notconstrained to be so.

Nearest-neighbor components

The SAR model formulation includes a regression parameter (sometimes denoted as a fixedeffect) for local spatial structure, so it offers a model-based alternative to the Geary test forthe presence of this structure. Any reduction in school- and district-level variance after theinclusion of Zikt indirectly estimates this local effect. Alternatively, we can identify a localspatial component directly by adding a random effect for variation in small neighborhoods.We operationalize this by partitioning the set of units into groups of about five nearestneighbors – thus nearby schools form a small pseudo-district, which we can include in anMLM model nested between schools and districts, at precisely the level of granularity thatwe desire.

Quantifying Spatial Components 7

The algorithm for building these pseudo-districts proceeds as follows: We construct alist, for each school, of its m nearest neighbors (m will be discussed shortly). Assignmentof each set of m + 1 schools to its own group is likely to result in overlapping groups, andalthough this would mirror the Geary and SAR weights quite closely, we prefer that thegroups be disjoint (this allows estimation using most MLM software). To ensure this, weassign groups consecutively, eliminating schools from consideration once they are assignedto a group. To minimize the number of singleton groups, we start with schools that are leastoften a nearest neighbor of another school. We follow this with a pass through the singletongroups, assigning them instead to a group with a smaller number of schools. Reassignmentsare made in a deterministic fashion. We experimented with several values of m, and foundthat the overlap reduces the average group size by about one school, so that m = 6 ensuresthat the median pseudo-district size is six (a school and its five nearest neighbors). Usingthis assignment mechanism, 82% of our schools are assigned to a group of size five, six orseven.

Our proximity-based grouping factor is smaller than districts, so it reflects local infor-mation in an intuitive manner. These pseudo-districts also cross district boundaries, whichallows us to pick up neighborhood effects. By including them (nested) in an MLM frame-work, we can evaluate their relative contribution to the overall variation. We simply add aterm (and an index) to our model:

Yijkt = β0 + β1t + β2t2 + αi + γj + δk + εijkt, (5)

where γj ∼ N(0, σ2γ) is the random effect associated with the jth pseudo-district, all other

stochastic components are as in the prior models and all are assumed independent of eachother. The variance components for this model, (σ2

α, σ2γ, σ2

δ , σ2ε), form a partition that

reflects the relative contribution of school, local (neighborhood), district, and residual vari-ation on test scores, net of the mean trend. We note that district effects may reflect somecoarse neighborhood effects as well. We fit the models using restricted maximum likelihoodestimation (REML) with the R (Ihaka and Gentleman, 1996) package LMER.

4. Additional predictors

When we add a predictor to a MLM, we often consider the level at which we expect it tohave an effect. If something is measured only at the district level, then it will not varyacross schools within districts, and thus it can be expected only to affect (or explain) thedistrict effects. Of course, all effects are estimated simultaneously, so adjustments made atthe district level may impact the effects of other predictors, upon which no matter whatlevel they operate. The situation is more complex with school (or unit) level predictors, inthat the aggregate level of these, say for district can be expected to have some relationshipto the district level effects as well as to the units within them. Allison (2005) and Neuhausand Kalbfleisch (1998) make this distinction as a between- versus within-cluster effect. Theterm sometimes used with MLMs is group-mean centering (see Bryk and Raudenbush, 2002and Enders and Tofighi, 2007). The idea is that the mean of a predictor, such as poverty,for a cluster, such as district, can influence the outcome for the entire district, and thusits random effect. This is the contextual (or between) effect of a predictor – being locatedin an overall poor district might explain why the outcomes in that district tend to below. Including district-level aggregate values of a predictor in a model thus captures howdifferences between district levels of poverty relate to the average outcomes for that district.

8 Scott et al.

They are thus likely to reduce the variance in district random effects. At the school-level,deviation from district-level means can be computed, and these reflect the impact of relativedifference on the outcome (a ‘within’ estimator). So within a poor district, do relativelybetter-off schools perform better or worse? For a single predictor x, we write x̄k for themean level of x in district k (we take this over all time points) and ∆xijkt for unit i indistrict k’s deviation xijkt − x̄k at time t and include both predictors in any model withcovariate effects.

Our model is thus extended to include one covariate and its corresponding group-leveleffect as follows:

Yijkt = β0 + β1t + β2t2 + βx;D x̄k + βx;S∆xijkt + αi + γj + δk + εijkt, (6)

with covariate terms defined as above. As Allison (2005) mentions, this can be understoodas a form of group-mean centering, common to the MLM literature; it also results in thefixed effects estimator for βx;S with respect to district, since the level of xijkt has beencleared out of the ∆xijkt term, and what remains is orthogonal to any district-constantpredictor. An advantage to this approach is that it allows one to compare between andwithin coefficients in a manner analogous to the Hausman test (Hausman, 1978; Allison,2005). The extent to which we clear out such between and within effects is not central tothe spatial aspects of the process, but it is directly applicable to the multi-level aspects ofit.

In our application, we will add a time-varying measure of poverty in each school popu-lation and then two indicators for the racial mix of the school population. We assess howthese predictors change the variance components associated with our three model types,which are MLMs with no spatial effects, the SAR model based on five nearest neighborsand the local random effects model based on nearest neighbor pseudo-districts. We evaluatethe presence or absence of residual spatial correlation in all models and examine maps thatreflect some of the more interesting variance components.

5. Application to NYC school test scores

Baseline model

Our baseline MLM includes a quadratic mean time trend and random effects for districtsand schools. The variance components for the latter two terms are estimated to be 167 and153, respectively, while the residual variance is estimated to be 66. We have ‘explained’ alarge portion of test score variation as district-level, which as we saw from figure 1 representscoarse spatial information as a proxy for a host of inputs. Differences between schools in adistrict exhibit variation of the same order of magnitude. This variance partition suggeststhat we may be able to find district- or school-level predictors that could ‘explain’ differencesbetween these units. Before we do that, however, we quantify the residual spatial variationthat we have not adequately captured.

We determine whether the residuals from nearby schools are unexpectedly correlatedwith each other using the Geary test, and this requires us to establish the null distributionof the test statistic. There are competing null distributions to use. A global test would positthat any score (or residual) can occur anywhere in the five boroughs of NYC. An objectionto this could be that it is likely to make our test too sensitive, as one or two districts couldbe spatially structured enough to set off the alarm for the whole city. A more conservativeapproach would conduct the Geary test within each district, and significant within district

Quantifying Spatial Components 9

spatial structure would be identified. This is a bit too conservative, because it will notdetect patterns that span two districts unless the number of schools affected is large enoughto be identified in both. Our solution is to let each test inform the other.

We apply the Geary test to the median school-level residual to smooth out any shortterm fluctuation. For our baseline model, the city-wide Geary test is significant (p =.011),while it is significant at the 0.05 level in 3 of the 32 districts (8, 14 and 16). While thereis a potential multiple comparisons problem in these 32 tests, when we reran them withan alpha level of 0.10, 6 districts were significantly residually spatially correlated, so thereis some reasonably robust evidence of structure. However, these findings also demonstratethe sensitivity of the test, and surprisingly isolated residual spatial correlation after trend,school and district effects are controlled.

The advantage to the MLM framework in the context of spatially situated data is thatthe variance components can literally be mapped. Figure 1 represents the raw score forone year, but it could just as easily be produced to contain all six years of scores and itwould look very similar. We decompose those scores into district, school and residual effectsestimated as BLUPs, with the residual taken as the median across years, and map thesein figures 2 through 4, respectively. The predominantly smooth changes as we move fromdistrict to district are depicted in figure 2; a bit less smooth, but fairly strongly patternedare the school effects in figure 3, which clearly sometimes suggest similar effects crossingdistrict boundaries. The Geary test, applied to those school effects is significant citywide(p < .0001) and significant at the .05 level in 13 of 32 districts. Our goal in the nextsubsection is to quantify further to what extent these school effects may be identified aslocal spatial structure. Note that the residuals depicted in figure 4 show very little structure,as confirmed by the district-specific Geary test. Perhaps District 8 contains the clearestresidual spatial structure, with better-performing schools located to the northeast.

[Figures 2 - 4 here]

Motivated by the structure identified in figure 3 and its corresponding Geary test, wequantify the local correlation using a SAR model. These are operationalized here as ad-ditional population-level effects based on the average of the PPE scores for the five ge-ographically nearest neighbor schools. Fitting that model shifted the variance decompo-sition, perhaps surprisingly, mostly as a reduction in district-level differences, which areestimated under this SAR model to have variance 61, while school-level variation drops asmall amount to 143, and residual variation is 63. The coefficient on the spatially-laggedPPE is 0.43 (p<.0001), suggesting that nearby test scores follow each other somewhatclosely. The large drop in district-level variation (over 100 units) suggests that some combi-nation of local and coarse spatial variation may be modeled quite effectively through spatiallags. In fact, the SAR model-based predictions are quite smooth; in figure 5 we plot thesepopulation-level predictions plus the corresponding district effects (omitting school effects),

β̂0 + β̂1t+ β̂2t2+ ρ̂∑

l wilYlkt + δ̂k (subscript j is dropped as no pseudo-districts are includedin this model), and find that this captures most of what we might call the spatial-signal (asopposed to school-level noise) in the data. This makes sense, because this additive modelusing spatially lagged predictors incorporates local information in a manner quite similarto models used for scatterplot smoothing (see Simonoff, 1996).

[Figure 5 here]

Finally, note that the MLM paradigm allows one to model random returns to spatiallylagged outcomes. When we augment the SAR model to include district-level variation in

10 Scott et al.

the lagged PPE coefficient, rather than a fixed effect of 0.43, we find a 95% confidenceregion for the potential spatial effects to be (0.19, 0.62), so the strength of local similarityvaries greatly. One could imagine developing the SAR paradigm further by including severalgrades of spatial lag (closer and further) and noting how these diminish in importance withdistance.

To summarize the findings so far, we have evidence, through the Geary test, that nearbyschool effects are correlated, and we have a measure of that correlation using the SARmodel. The latter utilizes fixed, or population-level effects to capture variation, and thereis some tension between using fixed or random effects to capture the same information; Jones(1990) discusses this issue in the longitudinal data setting. Our third class of models offers aconvenient middle ground: we include random effects for local pseudo-districts, constructedin a deterministic algorithm from nearest neighbors. The model was introduced in section 3.When fit to our school data, we find that the variance component for districts is estimatedto be 158 (a small drop), while it is 127 for schools (a bit larger drop). Residual variationremains steady at 66. The “missing” variation is captured in the pseudo-districts, witha component estimated to be 30. This is reasonably small in comparison to the othercomponents, but nearly half the size of the residual variation, suggesting that a small butimportant portion of the variation in school performance is locally spatial – the bulk ofthe variation remains at the school or district level (or unknown). We plot these local,nearest neighbor, components in figure 6. Note that while smaller in magnitude, theseeffects seem to be pick up within and across district variation quite well. So while smallin magnitude, there is still significant structure (p< 0.0001 for the LRT comparing thebaseline to the pseudo-district model). This structure corresponds to our intuition and isconfirmed in several different ways with the Geary tests and SAR model. The advantage tothe pseudo-district approach is that it resides in the MLM framework and admits a variancecomponents decomposition with district, local and school effects.

[Figure 6 here]

Poverty controls

We add a school-level poverty control, percent of students eligible for free or reduced-pricelunch (PFL), which we expect will “explain” some of the school-level variation. To theextent that there is aggregate district-level variation in poverty, it will explain those effectsas well. We formally explored this by including a district average for poverty along withschool-level deviations from that mean as suggested by Neuhaus and Kalbfleisch (1998).The first model we fit is:

Yijkt = β0 + β1t + β2t2 + βD;PFLPFLk + βS;PFL∆PFLijkt + αi + δk + εijkt. (7)

Note that we include subscript j because pseudo-districts may be included in some model,and this is how these terms were first introduced. This is our baseline model, with schooland district poverty controls, but no local spatial effects. We found it useful to includea district-level, random return to poverty: some districts are more or less responsive (interms of PPE test scores) to the relative poverty levels in their schools (p<0.0001 for LRTcomparing fixed returns to random returns). The model then becomes:

Yijkt = β0 + β1t + β2t2 + βD;PFLPFLk + (βS;PFL + φk)∆PFLijkt + αi + δk + εijkt, (8)

Quantifying Spatial Components 11

where φk is a district-specific adjustment to the return to PFL. We take (δk, φk)′ ∼ N2(~0, Σ),

where the covariance matrix Σ =

[

σ2δ σδ,φ

σδ,φ σ2φ

]

is unconstrained. In the baseline model,

the MLEs for β̂S;PFL and σ̂φ are -0.14 and 0.28, respectively, so a 95% confidence region for

β̂S;PFL + φ̂k covers the range (−0.68, 0.40), indicating both positive and negative district-

level associations with relative poverty. The MLE for β̂D;PFL is -0.68, which suggests thatdistrict-level differences have a greater impact than those at the school level. Both povertyfixed effects predictors are significant at the 0.05 level or better.

Introducing these predictors does change the variance components partition in fairlypredictable ways, but there are some challenges to incorporating the random return to PFLin the partition. To see this, we construct the total variance at the unit level based on theabove equation as a function of ∆PFLijkt, which we shorthand as ∆ijkt:

V (∆ijkt) = σ2α + σ2

δ + ∆2ijktσ

2φ + 2∆ijktσδ,φ + σ2

ε . (9)

The terms (σ2α, σ2

ε) represent school and unexplained variation, respectively. We can com-pute the average additional variation induced by district effects and district-level differen-

tial returns to relative poverty as σ2δ +

σ2

φ

N

∑

ijkt ∆2ijkt +

2σδ,φ

N

∑

ijkt ∆ijkt = σ2δ + σ2

φ∆2ijkt +

2σδ,φ∆ijkt, where N is the total number of observations across time for all units (schools)in the study. Using this summary measure for districts, which now contains more localspatial variation, the variance partition becomes 51 for districts, 99 for schools, and 65unexplained. compared to the baseline model, we are explaining a substantial portion ofdistrict-level variation through our two fixed effects.

We fit two related models using the same structure as (9), adding the new predictorZijkt =

∑

l wilYlkt as in (4) or γj , as in (5). Qualitatively, the partition under povertycontrols is quite similar for these model classes. As before, the SAR model explains a greaterportion of district-level differences through fixed, spatially lagged effects. The district-levelvariance component is 29 for the SAR model, while it is 45 in the nearest neighbor model.This components are depicted graphically in Figure 7, panels 2 and 3; each panel presents thevariance partition from fitting the no spatial component, SAR and nearest neighbor models,respectively. Within each panel, the baseline model is compared to models with additionalcovariate controls, the first being the addition of poverty, denoted as “+Poverty.” Nearestneighbor effects (panel 3) have dropped to 15 under poverty controls, and the school effectvariance is a bit smaller than that of the non-spatial model (2). Furthermore, residualspatial correlation appears to have been eliminated when evaluated city-wide all of ourmodels (under these poverty controls).

[Figure 7 here]

Race controls

We have measures of race and ethnicity aggregated to the school level, and include two inour next set of models: percent Asian and percent White (non-Latino). Interpreting theseeffects is complicated by the multiple processes they probably represent (e.g., residentialsorting, socioeconomic status). However, from an MLM perspective, noting which variancecomponents change after their addition is informative and may aid in the interpretation. Thetwo measures will be shorthanded as PAS and PWH and district-mean-corrected versions of

12 Scott et al.

these will be included along with their district-mean counterparts. The non-spatial modelthat includes all of our controls is thus:

Yijkt = β0 + β1t + β2t2 + βD;PFLPFLk + (βS;PFL + φk)∆PFLijkt + βD;PASPASk +

βS;PAS∆PASijkt + βD;PWH PWHk + βS;PWH ∆PWHijkt + αi + δk + εijkt,(10)

with all variance components as specified in (8).As depicted in the column labeled “+Race” in each panel in figure 7, the variance

component for school-level effects changes substantially and at about the same rate inall three models. It reaches parity, in terms of magnitude, with the unexplained portion ofvariation, which is just over 60. In all models, the between effects for race are significant andpositive; the within effects are nearly always significant and positive (only the SAR modellacks a significant effect for relative fraction white). However, race controls have shifted thelandscape enough that the within-district effect of poverty is no longer significant in anymodel. The district-level, random return to relative poverty is still warranted in all models(p<0.0001), so the importance of relative poverty persists in this manner. We assert nocausal interpretation to these findings; however, having explained a substantial portion ofthe originally documented spatially structured variation, poverty and race can be viewed ascrude, first approximations to the underlying unobservables that were previously identifiedas spatial. Of course, it is reasonable to assume that these are mere proxies for morenuanced unobserved covariates. Differences between SAR and our other model types arestill reflected in the variance partition (see figure 7). Much of the district-level variation iscaptured in the SAR coefficient ρ, leading to these differences, and even though ρ̂ drops inmagnitude with every additional control, it is still highly significant at this point, reinforcingthe idea that it captures several levels of spatial structure. While small, the nearest neighboreffect variance of 5, under these controls, is still significant (LRT p=0.048). These modelspartition the variance in a closely related but not identical manner.

6. Relevance to policy and discussion

We can use MLMs augmented with a “spatial lens” on outcomes in several different policy-relevant ways. First, we can document the extent of disparity that is spatial at multiplelevels. This research prevents a tractable approach to quantifying the local level of variation,the nearest neighbor approach, which may be buttressed by non-spatial MLMs followed byGeary tests on residuals or SAR models. Each has certain advantages, and may be chosenas the best descriptor of the process under different circumstances. Statistical correctionof standard errors to reflect residual spatial autocorrelation are important, and we providealternative modeling frameworks for so doing.

Once identified, changes in the variance components under covariate controls may informpolicy either by identifying important correlates of the spatial component or by revealingthe extent to which “space matters” at various levels of institutional control. The MLMrandom coefficient framework combined with a between versus within approach to covariatescan reveal differential responses to covariates. The extent to which space matters can becontrasted to the extent to which covariates matter at both local and broader levels.

We further note that these methods form an important complement to GIS methods, inthat statistical significance of what appear to be “hot spots” may be assessed. The methodscan also be used to confirm or deny the significance of patterning in stylized maps of covari-ates or outcomes. The methods, being model-based, yield several contrasting estimates of

Quantifying Spatial Components 13

latent spatial components, at different levels of spatial granularity. There is great potentialin linking these latent effects with new measures of neighborhood characteristics. Lastly,these methods can be used to spatially smooth outcomes with or without controls, yieldingmaps that more accurately reflect spatial signal rather than noise.

With some minor adjustment to these models, they may be used to track whether thespatial components (or the partitioning in the presence of covariates) changes over time. Itis important to evaluate change after sweeping reform, such as the Children First initiativein NYC (New York City Department of Education, 2002), which seeks to increase equity byremoving administrative barriers that were regionally based. Our models allow us to testspecific hypotheses about the spatial nature of the current system, unadjusted or adjusted,and at various levels. For the period studied here, total variation peaks in AY00-01 whilethe mean is nearing 50% from below, which is consistent with this change in proportion.The corresponding variance partitions rise and fall in tandem, suggesting that at least forthis period, the system was not moving toward unadjusted spatial parity. However, a morenuanced understanding of changes in the role of covariates over time could be instructive.

We have demonstrated an important perspective gained by quantifying the spatial com-ponent of variation in the evaluation of school disparity. Multi-level models, their spatialextensions and non-parametric tests provide us with an ample set of tools to meaningfullydecompose outcome variation. By employing models of this type, we provide the researcherwith the ability to test new and important hypotheses about the role of geography and usethe results to inform policy.

14 Scott et al.

References

[1] Allison, P.D. (2005). Fixed Effects Regression Methods for Longitudinal Data UsingSAS. Cary, NC: SAS Institute Inc.

[2] Anselin, Luc and Bera, Anil (1998). Spatial dependence in linear regression modelswith an introduction to spatial econometrics. In Amman Ullah and David Giles (eds.),Handbook of Applied Economic Statistics. New York: Marcel Dekker: 237–289.

[3] Brooks-Gunn, J., Duncan, G.J., and Aber, J.L. (eds.) (1997). Neighborhood poverty:Context and consequences for children. New York: Russell Sage Foundation.

[4] Bryk, A.S. and Raudenbush, S.W. (2002). Hierarchical Linear Models: Applicationsand Data Analysis Methods Thousand Oaks, CA: Sage.

[5] Cliff, A. and Ord, J.K. (1981). Spatial Processes: Models and Applications. London:Pion.

[6] Cressie, Noel A.C. (1986). Kriging Nonstationary Data. Journal of the American Sta-tistical Association, 81: 625-634.

[7] Cressie, Noel A.C. (1993). Statistics for spatial data. New York: Wiley.

[8] Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. New York:Chapman and Hall.

[9] Enders, C.K. and Tofighi, D. (2007). Centering predictor variables in cross-sectionalmultilevel models: A new look at an old issue. Psychological Methods, 12(2):121-138.

[11] Gelman, A. and Hill, J. (2007). Data Analysis Using Regression and Hierarchi-cal/Multilevel Models, New York: Chapman & Hall.

[11] Grimson R., Wang K., and Johnson, P. (1981). Searching for hierarchical clusters ofdisease: spatial patterns of sudden infant death syndrome. Social Science and Medicine,15D: 287-293.

[12] Haining, Robert (2003) Spatial data analysis: Theory and practice. Cambridge: Cam-bridge University Press.

[13] Hausman, J.A. (1978). Specification Tests in Econometrics. Econometrica, 46(6), 1251-1271.

[14] Ihaka R, Gentleman R. (1996). R: A language for data analysis and graphics. Journalof Computational and Graphical Statistics, 5: 299-314.

[15] Jencks, Christopher, and Susan E. Mayer (1990). The social consequences of growingup in a poor neighborhood. In Lynn, Laurence E. and Michael McGreary (eds.), Inner-City Poverty in the United States. Washington, DC: National Academies Press.

[16] Neuhaus, J.M. and Kalbfleisch, J.D. (1995). Between and within-cluster Covariateeffects in the analysis of clustered data. Biometrics, 54, 638-645.

[17] New York City Department of City Planning. (2006). City of Neighborhoods. Available:http://www.nyc.gov/html/dcp/html/neighbor/neigh.shtml

Quantifying Spatial Components 15

[18] New York City Department of Education (2002). Klein announces Children First: Anew agenda for public education in New York. Press release. Available:http://www.nycenet.edu/press/02-03/n36 03.htm

[19] Robinson, G.K. (1991). That BLUP is a good thing: the estimation of random effects.Statistical Science, 6(1): 15-32.

[20] Searle, Shayle R., Casella, George, and McCulloch, Charles E. (1992). Variance com-ponents. New York: Wiley.

[21] Simonoff, J.S. (1996). Smoothing Methods In Statistics. New York: Springer.

[22] Small, M.L. and Newman, K. (2001). Urban poverty after the truly disadvantaged: Therediscovery of the family, the neighborhood, and culture. Annual Review of Sociology,27:23-45.

[23] Wilson, William Julius. (1987). The truly disadvantaged: The inner city, the under-class, and public policy. Chicago: University of Chicago Press.

Figure 1. PPE raw scores: AY03−04

Longitude (W)

Latit

ude

(N)

40.5

40.6

40.7

40.8

40.9

74.2 74.1 74 73.9 73.8 73.7

●●●

●●●

●●●

●●

●

●●●●●

●●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●●

●

●

●

●●

●●●

●● ●●●

●● ●

●

●

●●

●

●●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●●

●●

●

●

●

●

●●

●

●●

●

●

●●

● ●

●

● ● ●●●

●●●●

●●

●●

●

●

●

●●●

●

● ●

●

●●●

●● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●●●

●

●●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●●

●●●

●●

●●

●●

●●

● ●●

●

●●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●●

●

●●

●

● ●

●

●●

●●

●

●

●●

●●

●● ●

●

●●

●●

●●

●●

●

●●

●

●

●

●●●

●●

●●

●

●

●

●

●

●

●●●

●●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●●●●●

●

●●●●

●●

●●

●●●

●

●●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

● ●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

18

19

20

15

22

21

32

23

14

16

17

13

1

6

2

43

5

10

8

7

11

9 12

30

29

2625

24

27

28

31

Manhattan

The Bronx

Queens

Brooklyn

Staten Island

89847872666054484237312519

Figure 2. District PPE Effects

Longitude (W)

Latit

ude

(N)

40.5

40.6

40.7

40.8

40.9

74.2 74.1 74 73.9 73.8 73.7

●●●

●●●

●●●

●●

●

●●●●●

●●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●●

●●●

●● ●●●

●● ●

●

●

●●

●

●●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●●

●

●●

●

●

●●

● ●

●

● ● ●●●

●●●●

●●

●●

●

●

●

●●●

●

● ●

●

●●●

●● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●●

●●●

●●

●●

●●

●●

● ●●

●

●●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●●

●

●

●

●

●

●●

●

●●

●

● ●

●

●●

●●

●

●

●●

●●

●● ●

●

●●

●●

●●

●●

●

●●

●

●

●

●●●

●●

●●

●

●

●

●

●

●●

●●●

●●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●●●●●

●

●●●●

●●

●●

●●●

●

●●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

●

● ●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

18

19

20

15

22

21

32

23

14

16

17

13

1

6

2

43

5

10

8

7

11

9 12

30

29

2625

24

27

28

31

Manhattan

The Bronx

Queens

Brooklyn

Staten Island

3632282319151172−2−6−10−15

Figure 3. School PPE Effects (net of District)

Longitude (W)

Latit

ude

(N)

40.5

40.6

40.7

40.8

40.9

74.2 74.1 74 73.9 73.8 73.7

●●●

●●●

●●●

●●

●

●●●●●

●●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●●

●●●

●● ●●●

●● ●

●

●

●●

●

●●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●●

●

●●

●

●

●●

● ●

●

● ● ●●●

●●●●

●●

●●

●

●

●

●●●

●

● ●

●

●●●

●● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●●

●●●

●●

●●

●●

●●

● ●●

●

●●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●●

●

●

●

●

●

●●

●

●●

●

● ●

●

●●

●●

●

●

●●

●●

●● ●

●

●●

●●

●●

●●

●

●●

●

●

●

●●●

●●

●●

●

●

●

●

●

●●

●●●

●●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●●●●●

●

●●●●

●●

●●

●●●

●

●●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

●

● ●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

18

19

20

15

22

21

32

23

14

16

17

13

1

6

2

43

5

10

8

7

11

9 12

30

29

2625

24

27

28

31

Manhattan

The Bronx

Queens

Brooklyn

Staten Island

252117131062−2−5−9−13−17−21

Figure 4. Median PPE Residuals (net of district and school effects)

Longitude (W)

Latit

ude

(N)

40.5

40.6

40.7

40.8

40.9

74.2 74.1 74 73.9 73.8 73.7

●●●

●●●

●●●

●●

●

●●●●●

●●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●●

●●●

●● ●●●

●● ●

●

●

●●

●

●●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●●

●

●●

●

●

●●

● ●

●

● ● ●●●

●●●●

●●

●●

●

●

●

●●●

●

● ●

●

●●●

●● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●●

●●●

●●

●●

●●

●●

● ●●

●

●●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●●

●

●

●

●

●

●●

●

●●

●

● ●

●

●●

●●

●

●

●●

●●

●● ●

●

●●

●●

●●

●●

●

●●

●

●

●

●●●

●●

●●

●

●

●

●

●

●●

●●●

●●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●●●●●

●

●●●●

●●

●●

●●●

●

●●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

●

● ●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

18

19

20

15

22

21

32

23

14

16

17

13

1

6

2

43

5

10

8

7

11

9 12

30

29

2625

24

27

28

31

Manhattan

The Bronx

Queens

Brooklyn

Staten Island

4332110−1−1−2−3−3−4

Figure 5. SAR Model−Based PPE Predictions (purged of school effects)

Longitude (W)

Latit

ude

(N)

40.5

40.6

40.7

40.8

40.9

74.2 74.1 74 73.9 73.8 73.7

●●

●●

●●

●●

●

●● ●

●●

●●

●●

●

●

●●

●●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●●●

●

● ●●

●

●

●●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ● ●●

●

●

●

●

●●

●

●

●

●

●●

●●

●

● ●●

●

●

●●●

●

●

●

●

●

●●

●

●

● ●

●

●●●

●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●●

●

●

●

●

●

●●

●●●

●●

●

●●

●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●

●●

●

● ●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

● ●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●●●

●●

●

●●●●

●●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

● ●

●

● ●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●

●●

●

●●

●

●●

●●

● ●

●

●

●●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●●

●

●

●●●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

● ●●

●

● ●●●

●

●

●

●

●

●●

●

●

●●

●

●●

●

●● ●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●●

●●●

●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●●

●●

●

● ●

●

●●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●●

●

●●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

● ●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●●

●● ●

●●

●

●●

●

●●

●●

● ●

●

●

●●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●●

●●

●●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●

●

● ●●

●

●●

●● ●

●

●

●

●

●

●●

●

● ●

●

●●●

●● ●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●●

●●

●●●

●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●●

●●

●

●

●●

●●

●● ●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

● ●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●●

●

●

●●

●●

●●

●● ●

●●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

● ●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●●

●●●

●● ●

●●

●●

● ●

●

●

●●

●●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●

●● ●●

●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

● ●●

●

●

●

●

● ●

●

●

●

●

●● ●

●

● ●

●

●●●

●

● ●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●

●

●●

●

●●

●

●●

●

●●

●

●

●●

●●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●●

●●

●

●

●●

●●

●

●●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●●

●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●

●

●

●

●●●

●●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●●

●

●

●●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●●●

●

● ●

●●

● ●

● ●

●

●

●●

●●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●● ●

●

●●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●●●

●

● ●●● ●

●

●

●

●

●

●●

●

● ●

●

●●●

●

● ●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●●

●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●●

● ●●

●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●●

●

●●

●●

●●

●●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●

●

●●

●●

●●

●

●●

● ●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●●

●●●

●

●●

●●

●●

● ●

●

●

●●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●●

●● ●

●● ●●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●

●

● ●●

●

●

●●●

●

●

●

●

●

●

● ●

●

● ●

●

●● ●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●●

●●●

●

●●

●●

●

●

● ●

●

●

●●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●●

●

●●

●

●

●

●●

●

●

●●

●●

●●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●●●

●

●●

●●

●●

●

● ●● ●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●●

●

●

●●

●

●

● ●

●● ●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●●

●

●●

●

●●

● ●

● ●

●

●

●

●

●●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●

● ●

●● ●●●

●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●

●

● ● ●●

●●

●●● ●

●

●

●

●

●

●

● ●

●

● ●

●

●●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●●

●●

●

●

● ●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●

●

●

●●

●

● ●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

● ●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●●

●

●

●●

●●

●●

●●

●●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

● ●●

●

●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●●● ●

●●●●●

●●●

● ●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●●

●● ●

●●

●●●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●●

●●

●

●

●

●

●●

●

●

●

●

●●

●●

●

● ● ●●●

●

●●● ●

●

●●

●

●

●●

●

●

●●

●

●●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

● ●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

● ●

●

●

●

●●

●●

●

●●

●

●

●

●●

●●

●

●●

●

●●

●

●

●●

●●

●

●

●

●●

●●●

●

● ●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●●

●●

●

●●

●

●

●

●●

●

●

●●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●

●●

●

●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●●

●

●

●

●

●●

●●

●●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●●

●

●●

●●

●

● ●

● ●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●●

●● ●

●

● ●●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

● ● ●●

●

●●

●● ●

●

●

●

●

●

●● ●

●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●●

●●

●●

●

●●

●●

●

●

●●

● ●

●

●

●●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●

●

●●

●

●

●

●

●●

●

● ●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●●●

●

●●

●●

●●

●●

●

●●

●

●

●● ●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●●● ●

●●●

●● ●

●●

● ●

●●

●

●

●●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●● ●

●

● ●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●● ●

●

● ● ●●

●

●

●●● ●

●

●

●

●

●

●● ●

●

●●

●

●●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●●

●

●●

●●

●●

●

●●

●

● ●

●

●

●●

●●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●●

●●

●

● ●

●

●●

●●

●

●

●●

●●

●

● ●

●

●●

●●

●●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

● ●●

●● ●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●●●

●

●●●

●●●

●●●

● ●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●●

●●●

●●

●

●●

● ●

● ●

●

●

●●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●● ●

●

● ●●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●● ●

●

●

●●

●● ●

●

●

●

●

●

●● ●

●

● ●

●

●●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●

●

●●

●●

●●

●

●

●●

●●

●

●

●●

●●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●●

●●

●

● ●

●

●●

●●

●

●

●●

●●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●

●●

●●

●● ●

●

●●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●●

●●●

●●

●

●●

●●

●●

●

●

●●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●●●

●

● ●●●

●

●

●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

● ●●

●●

●

●

●

●●

●

●

●

●

●

●

●●

●

● ● ●●

●

●●

●● ●

●

●●

●

●

●

●●

●

● ●

●

●● ●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●●

●

●

●

●

●

●●

●

●●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●●

●●

●

●

●

●

●●

●● ●

●

●●

●●

●●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●

●

●

●●●●

●●

●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●●

●

●

●●

●●●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●●

●●●

●●

●

●

●●● ●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●●

●●

●●●

●

● ●●

●

●

● ●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

● ● ●●●

●

●●●

●

●

●●

●

●

●

●

●●

●

●●

●

●● ●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●

●

●●

●●

●●

●

●

●

●

● ●

●

●

●●

●●

●

●

●

●●

●

●●

●

● ●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●●

●

● ●

●

●

●

●●

●

●

●●

●●

●● ●

●

●●

●●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

● ●

●

●● ●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●●

●

●

●●●

●

●

●● ●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●

●

●

●

● ●

●●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●●

●●●

●●

●

●

●●

●●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

●● ●●

●

●

● ●●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

● ●●●

●

●●●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●●●

●

● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●●

●

●

●

●

●

●●

●●●

●●

●

●●

●●

●●

●

●

●

●

● ●

●

●

●●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●●

●● ●

●

●

●

●●

●●

●●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

● ●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●

●

●

●●

●●

●

●●

●●

●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

●

● ●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●●●

●

●●

●

●●● ●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●●

●

●●

●

●●●

●

●

● ●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●●

●

● ● ●●

●

●

●● ●●

●

●

●

●

●

●

●● ●

●

● ●

●

●●●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●●

●

●

●

●

●

●●

●●●

●●

●●

●

●

●

●●

●●

●●

● ●

●

●

●●

●●

●

●

●

●

●

● ●●

●

● ●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●

●●●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●●

●●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

● ●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

● ●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●●

●

●●

●●

●

●

●●

●

● ●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

● ●

●

●

●

● ●

●●

●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●●

●

●

●

●

●●

●

●

●

● ●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●●●

●●

●

●

●●

● ●

● ●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●●

●●●

●● ●●

●

●

● ●

●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●●●

●

●

●●● ●

●

● ● ●●

●

●

●●●

●

●

●

●●

●

●

●● ●

●

● ●

●

●●●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●

●●

●

●●

●●

●●

●

●

● ●

●

●

●●

●●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●●

●●

●

●●

●

●

●

● ●

●

●

●●

●●

●● ●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●●●

●

●

●●●

●●

●● ●

● ●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●●

●

●

●

●●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●●●

●●

●

●

●●●

●

● ●

●

●

●●

●●

●

●●

●

●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●●

●●●

●●

●●

●

●

● ●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●●

●

●●

●

●

●●

●●

●

● ● ●●

●

●●●●

●

●

●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●●

●

●

●● ●

●

●

●

●

●●

●

●

●

●

●

●●

●●●

●●

●

●●

● ●

●●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●

●

●●●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●

●

●●

●

●

●●

●●

●

● ●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

● ● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●

●

●

●●●

●

●●

●●

●

●●

●

●●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●● ●

●●●

●●

●

●

●●●

●

●●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●●

●

●

●●

●

●

● ●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●●

●

●●

●

●

●●

●●

●

● ● ●●●

●●

● ●●

●

●●

●

●

●

●●

●

●

● ●

●

●●●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●●

●●

●

●●

●●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●●

●

●●

●

● ●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●●

●●

●

●

●

●

●●

●● ●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●●

●

●● ●●

●

●

●●

●

●

●●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●●

●

●

●●●

●●

●●●

● ●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

●

● ●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●●● ●

●●

●

●●

●

●

●●

● ●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

●●

●●●

●

● ●●

●

●

● ●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ● ●

●

●

●

●

●

●●

●

●●

●

●

●●

●●

●

● ● ●●

●●

●●● ●●

●

●●

●

●

●● ●

●

● ●

●

●●

●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●●

●●

●

●

●

●●

●●

●●

●●

●

●

●●

●●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●●

●●

●

● ●

●

●

●

●

●

●

●

●●

●●

●

●●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●●●

●●●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●●

●

●●

●●

●●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

● ●

●

●

●

● ●

●●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●● ●

●●●

●

●●

●

●●

● ●

●●

●

●

●●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●● ●

●

●

●

●●

●●

●

●

● ●●●

●●

●

●

●

●●

●

●●●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

● ●●

●●

●

●

●

●●

●

●●

●

●

●●

● ●

●

● ●●●

●

●●

●● ●

●

●

●●

●

●

●

●

●●

●●

●

●● ●

●● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●

●

●●

●●

●

●

●

●

●

●

● ●

●

●

●●

●●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●●●

●

● ●

●

●

●

●

●

●

●

●●

●●

●

●●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●●●

●

●

●●

●

●

●

●

●

●●

● ●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

● ●

●●

●●

●

● ●●

●●

●

●●

●

●●

●

●●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●

●

●

●

● ●

●● ●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●●● ●

●●

●

●

●

●

●

●●

● ●

● ●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●●

●● ●

●

●

●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

●●

●●

●

●

●●

● ●

●

●

●●

●

●

●

●●

●

● ●

●

●● ●

●

● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

● ●

●

●

●

●●

●●

●●

●

●●

●

●

●

●●

●

●

●

●

● ●

●

●

●●

●●

●

●

●

● ●

●

●●

●

● ●

●

●

●

●

●

●

●

●●

●●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●●

●● ●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

● ●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

● ●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●●

●

●●

●●●●●

●● ●

● ●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

● ●

●

●

●

●●

●

●●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●●●

●●

●

●

●●

●●

● ●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

●●

●

●●●

●

●

● ●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●● ●

●

●

●

●●●

●

●

●

●●

●

●

●

●●

●

● ●

●

●●●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●●

●

●

●

●

●

●●

●●

●●

●●

●●

●

●

●●

●●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●●

●

● ●

●

●●

●

●

●

●

●●

●●

●

● ●

●

●●

●●

●●

●●

●

●●

●

●

●

●●●

●

●

●●

●

●

●

●

●

●●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●●

●

●

●

●●● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●●●

●

●●

●●

●

●

●●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

●

●●

●●

●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●

●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●●●

●●●

●●

●

●●

●

●

●●

●●

●●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●●

●●●

●●

●●

●

●

● ●●

●

●

● ●

●●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●●

●

●●

●

●

●●

●●

●

● ● ●●

●

●●

●●●

●

●●

●

●

●

●● ●

●

●●

●

●●

●

●

● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●●

●

●●

●●

●●

●●

●

●

●●

●

●

●●

●●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●●

●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●●●

●

●●

●

●●●

●●

●● ●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

●●

●

●

●

● ●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●● ●

●●●

●

●●

●

●●● ●

● ●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●●

●●

●●

●

●

● ●●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●●

●

● ●●●

●

●

●●●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●●●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●●

●

●

●

●

●●

●●

●

●

●

●

● ●●

●

●●

●●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●

●

●●

●

●

●●

●●

●

●●

●

●

●

●●

●●

●●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

● ●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●●

●

●

●●

●●●

●●

●● ●

●

●

●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●●

●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●●

●●

●

●

●●

●●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●●

●

●

● ●●●

●

● ●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●●

●

●

●●

●●

●

● ●●●

●●

●●● ●

●

●

●●

●

●

●

●●

●

● ●

●

●●●

●

● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●●●

●●

●

●●●

●●

●●

●

●●

●

● ●

●

●

●●

●●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●●

●

●

●●

●●

●● ●

●

●●

●●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

● ●●●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●●

●

●

●●●

●

●

●●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

● ●

●

●

●

● ●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●●

●

●

●●

● ●

● ●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●●

●

●●●

●

●

● ●●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●●

●●

●

●●

●●●

●

●

●●

●

●

●●

●

●

●●

●

●●●

●

● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●●●

●

●●

● ●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●●

●

●●

●

● ●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●●

●

●

●●

●●

●

● ●

●

●

●

●●

●●

●●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●●

●

●

● ●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●

●●

●●

●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

●

● ●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●●

●●●

●●

●

●

●●

● ●

●●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●●

●

●

●

●●

●●●

●

● ●●●

●

● ●●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●● ●

●

●

●

●●●●

●

●●

●

●

●

●● ●

●

● ●

●

●●●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●

●

●●

● ●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●●●

●

● ●

●

●

●

●●

●

●

●

●

●●

●

● ●

●

●●

●●

●●

●●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

● ●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●●●

●

●●●

●

●●

●

●●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

● ●

●

●

●

● ●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●●

●

●●●

●

●●

● ●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●●

●●●

●●

●●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●● ●

●

●

●

●

●

●●

●

●●

●

●

●●

●●

●

● ● ●●●

●●

●● ●

●

●

●●

●

●

●● ●

●

● ●

●

●●●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●●

●

●●

●

●

●●

●

●●

●

● ●

●

●

●●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●●

●

● ●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●●

●

●●●

●

●●

●

●

●

● ●

●

●●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●●●

●●

●

●

●●

●●

● ●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●●

●● ●

●

● ●●

●

●

● ●

●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●●

●

●

●

●

●●

●

●●

●

●

●●

●●

●

● ● ●●

●

●●●●

●

●

●●

●

●

●

●● ●

●

● ●

●

●●●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●●

●●

●

●

●

●●

●

●●

●

●●

●

●

●●

●●

●

●

●

●

●

● ●●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●

●●

●●

●

●●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

● ●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●●●

●

●

●

●●●●

●●

●●

●●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●

●

●

●

● ●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●●●

●●

●

●●●

● ●

●●

●

●

●●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

●

● ●●●

●

● ●●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

● ● ●●

●

●●●

● ●

●

●

●●

●

●

●●

●

●

● ●

●

●●●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●●

●●●

●●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●

● ●●

●

● ●

●

●

●

●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●●

●

●●

●●

●

●

●●

●●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●●

●

●

●●

●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●●

●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●

●

●

●●

●●

●●

●●

●● ●

●

●

●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●●

●

●

●●

●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●●

●●

●

●

●●

●

●●● ●

●●

●

●

●●

●●

●

● ●

●

●

●

●

●●

●●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●●

●

●●●

●

●

●●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●●

●

●●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●● ●

●

●

●

●●● ●

●

●●

●

●

●

●●

●

●

● ●

●

●●●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

● ●●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●

●

●●

●●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●●

●

●

●

●

●●

●●

●

●

●●

●●

●

● ●

●

●●

●

●

●●

● ●●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●● ●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●

●

●

●●●

●

●●

●●

●

● ●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

● ●

●

●

● ●

●

●●

● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●● ●

●●

●

●

●●

●●

●● ●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●● ●

●● ●●

●

●

● ●●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

● ● ●●●

●●

●● ●

●

●

●●

●

●

●

●●

●

● ●

●

●●●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

● ●●

●

●●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●

●

●

●

●

●

●●

●

●

●●

● ●

●

●

●●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●●

●●

●

● ●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●● ●

●●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●

●

●

● ●●●

●●

●● ●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

● ●

●

●

●●

●●

●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●

●●●

●●

●●

●●

●●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●●●

●● ●●●

●

● ●

●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●●

●

●●

●

●

●●

●●

●

● ● ●●●

● ●●

● ●

●

●

●

●

●

●

●● ●

●

●●

●

●●●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●●●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●

●●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●

● ●

●

●●

●

● ●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●

●

●●

●

●

●

●

●●

●

● ●

●

●●

●●

●●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

● ●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●●

●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●●

●●

●●

●● ●

● ●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

●●

●●

●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●●

●

●●

●

●

●●

●●

●●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●● ●

●

●

●●

●●●

●

● ●●

●

●

● ●

●

●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

● ●●●

●

●

●●●

●

●

●●

●

●

●

●

●

●

●

●●

●

●●●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

● ●

●

●

●

●●

●

●

●

●

●●

●

● ●

●

●

●

●

●●

●

●●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●●●

●

●●●●

●●

●●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

● ●●

●

●

●

●●

●

●

●

●

●

●

●

●●

●●

●

●

●●

●

●

●●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●●

●●●

●●

●

●

●●●●

●●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●●●●●●

●

●

● ●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●●

●

●●

●

●

●●

● ●

●

● ● ●●

●

●●

●●

●●

●●

●

●

●

●●

●

●

● ●

●

●●●

●

●●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●●

●●

●

●●

●

●●

●●

●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●

●

●●

●

●

●

●

●●

●● ●

●

●●

●●

●●

●●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

● ● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●

●

●

●

●●●●

●●

●●●

● ●

●

●●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●

●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●●●

●

●●

●

●●

●●

● ●

●

●

●●

●●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●●

●●●●

●

●

● ●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●●

●

●●

●

●

●

●

●●

●

● ● ●●

●

● ●●● ●

●

●

●●

●

●

●

● ●

●

● ●

●

●●●

●

● ●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●●●

●

●●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●●

●●

●

●

●

●●

●●

●●

●

●

●●

●●

●●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●●

●●●

●

●

●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●●

●

●

●●

●●

●

● ●

●

●

●

●●

●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

●

●

●● ●

●● ●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●●●●

●

●

●●●

●●

●● ●●

●●

●●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●

●

●

●●

●

●●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●●

●●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

18

19

20

15

22

21

32

23

14

16

17

13

1

6

2

43

5

10

8

7

11

9 12

30

29

2625

24

27

28

31

Manhattan

The Bronx

Queens

Brooklyn

Staten Island

78736964595550454136312722

Figure 6. Nearest neighbor effect (net of district and school effects)

Longitude (W)

Latit

ude

(N)

40.5

40.6

40.7

40.8

40.9

74.2 74.1 74 73.9 73.8 73.7

●●●

●●●

●●●

●●

●

●●●●●

●●

●

●

●●

●●

●

●●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●●

●●●

●● ●●●

●● ●

●

●

●●

●

●●●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●●

●●

●

●

●

●

●●

●

●●

●

●

●●

● ●

●

● ● ●●●

●●●●

●●

●●

●

●

●

●●●

●

● ●

●

●●●

●● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

● ●●

●

●

●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●●

●●

●

●

●

●●

●●●

●●

●●●

●●

●●

●●

●●

● ●●

●

●●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●●

●

●

●

●

●

●●

●

●●

●

● ●

●

●●

●●

●

●

●●

●●

●● ●

●

●●

●●

●●

●●

●

●●

●

●

●

●●●

●●

●●

●

●

●

●

●

●●

●●●

●●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●●

●

●

●

● ●● ●

●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●●

●●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●●●●●

●

●●●●

●●

●●

●●●

●

●●

●●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

● ●

●●

●

●

● ●

●

●

●

● ●

●●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●●

●

●

●●●

●

●

●

●

●

●●

●

●

●

●●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●●

●●●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●●

●

●

18

19

20

15

22

21

32

23

14

16

17

13

1

6

2

43

5

10

8

7

11

9 12

30

29

2625

24

27

28

31

Manhattan

The Bronx

Queens

Brooklyn

Staten Island

9765321−0−2−3−4−6−7

No Local Spatial Effects

Var

ianc

e

050

100

150

Baseline +Poverty +Race

Unexplained

School

District

Spatial Auto−Regression

Var

ianc

e

050

100

150

Baseline +Poverty +Race

Unexplained

School

District

Nearest−Neighbors Effects

Var

ianc

e

050

100

150

Baseline +Poverty +Race

Unexplained

School

District

Neighbors

Figure 7. Variance Components: various model types and predictors